[Paper Review] Power and Laurent series in the algebra of complex quaternions

T. S. Kuzmenko|arXiv (Cornell University)|Mar 17, 2015

Algebraic and Geometric Analysis3 citations

TL;DR

This paper establishes Taylor and Laurent series expansions for $G$-monogenic mappings with values in the algebra of complex quaternions, introducing a novel framework based on a specific idempotent basis. It classifies isolated singularities of these mappings as removable, poles, or essential, extending classical complex analysis concepts to the non-commutative quaternionic setting using functional decomposition into complex-valued components.

ABSTRACT

Taylor's and Laurent's expansions of $G$-monogenic mappings taking values in the algebra of complex quaternion are obtained and singularities of these mappings are classified.

Motivation & Objective

• To extend classical complex analysis results—specifically Taylor and Laurent series expansions—to $G$-monogenic mappings in the non-commutative algebra of complex quaternions.
• To define and classify isolated singularities of $G$-monogenic mappings in terms of the behavior of their component functions in the complex plane.
• To provide a constructive framework for series expansions of $G$-monogenic mappings using a specific idempotent basis $\{e_1, e_2, e_3, e_4\}$, enabling decomposition into holomorphic components.
• To generalize the concept of isolated singularities (removable, pole, essential) from complex analysis to the context of quaternionic mappings via the functional components $F_1, F_2, F_3, F_4$.

Proposed method

• Represent $G$-monogenic mappings $\Phi: \Omega_\zeta \to \mathbb{H}(\mathbb{C})$ using an idempotent basis $\{e_1, e_2, e_3, e_4\}$, where $\zeta = \xi_1 e_1 + \xi_2 e_2$ with $\xi_1 = f_1(\zeta)$, $\xi_2 = f_2(\zeta)$, and $\|\zeta\| = \sqrt{|\xi_1|^2 + |\xi_2|^2}$.
• Decompose the mapping $\Phi(\zeta) = F_1(\xi_1, \xi_2)e_1 + F_2(\xi_1, \xi_2)e_2 + F_3(\xi_1, \xi_2)e_3 + F_4(\xi_1, \xi_2)e_4$, reducing the problem to analyzing holomorphic functions $F_k: \mathbb{C}^2 \to \mathbb{C}$.
• Derive Taylor and Laurent series expansions for $\Phi$ by applying classical complex analysis techniques to the component functions $F_k$, leveraging the fact that $\Phi$ is $G$-monogenic if and only if its components satisfy certain analyticity conditions.
• Use the resolvent identity $(t - \zeta)^{-1} = \frac{1}{t - \xi_1}e_1 + \frac{1}{t - \xi_2}e_2$ to identify the singular sets as lines $L^1, L^2$ in $\mathbb{R}^3$, corresponding to $\xi_1 = \xi_{10}$ and $\xi_2 = \xi_{20}$.
• Classify singularities of $\Phi$ by analyzing the behavior of $F_k$ at $\xi_{10}$ and $\xi_{20}$: a point $\zeta_0$ is removable if all $F_k$ have finite limits, a pole if at least one $F_k$ has an infinite limit, and essential if no limit exists.
• Define the singular set $\widetilde{\mathcal{K}}_\zeta^0 \cap \{\zeta_0 + e^* : e^* \in L_\zeta^1 \cup L_\zeta^2\}$ as the locus of non-removable singularities, and prove that if the principal part of the Laurent series has infinitely many non-zero terms, then at least one $F_k$ has an essential singularity.

Experimental results

Research questions

• RQ1 How can Taylor and Laurent series expansions be generalized to $G$-monogenic mappings with values in the algebra of complex quaternions?
• RQ2 What is the structure of the singular set of a $G$-monogenic mapping, and how does it relate to the singularities of its component functions in the complex plane?
• RQ3 How do the concepts of removable, pole, and essential singularities from complex analysis extend to $G$-monogenic mappings in the quaternionic setting?
• RQ4 Under what conditions does a $G$-monogenic mapping fail to have a finite or infinite limit at a point, and how is this related to the behavior of its component functions?

Key findings

• A right-$G$-monogenic mapping $\Phi$ admits a Taylor series expansion $\Phi(\zeta) = \sum_{n=0}^\infty (\zeta - \zeta_0)^n p_n$ in a ball $B(\zeta_0, R)$, where $R = \min_{\zeta \in \partial \Omega_\zeta} \|\zeta - \zeta_0\|$, with coefficients $p_n = \frac{1}{2\pi i} \int_{\gamma_\zeta} (\tau - \zeta_0)^{-n-1} d\tau \, \Phi(\tau)$.
• A $G$-monogenic mapping admits a Laurent series expansion in a punctured neighborhood of $\zeta_0$ if and only if its component functions $F_k$ admit Laurent expansions in $\xi_1$ and $\xi_2$, with the principal part determined by the singularities of these components.
• An isolated singularity $\zeta_0$ of $\Phi$ is removable if and only if all component functions $F_k$ have finite limits at $\xi_{10}$ and $\xi_{20}$; otherwise, if at least one $F_k$ has an infinite limit, $\zeta_0$ is a pole.
• If the principal part of the Laurent series of $\Phi$ contains infinitely many non-zero terms, then at least one of the component functions $F_k$ has an essential singularity at $\xi_{10}$ or $\xi_{20}$, implying $\Phi$ has no finite or infinite limit at $\zeta_0$.
• The set of non-removable singularities of $\Phi$ is contained in $\widetilde{\mathcal{K}}_\zeta^0 \cap \{\zeta_0 + e^* : e^* \in L_\zeta^1 \cup L_\zeta^2\}$, where $L_\zeta^1$ and $L_\zeta^2$ are the lifts of the lines $L^1$ and $L^2$ in $\mathbb{R}^3$ to $E_3$.
• If $F_1 \equiv 0$, $F_3 \equiv 0$, $F_4 \equiv 0$, and $F_2$ has an essential singularity at $\xi_{20}$, then $\Phi$ has no finite or infinite limit at $\zeta_0 + e^*$ for $e^* \in L_\zeta^2$, confirming the existence of essential singularities in the quaternionic setting.